Two days later, Kelly surveyed the same 13 classmates and found that none of them had been given math homework since she last surveyed them.



By how much does the mean of Kelly’s second data set change in comparison with the mean of the data set in her original survey? Explain how to determine the change in the means without calculating the mean of either data set

The annual interest rate the third party will receive for the investment is approximately 7.7%.

After buying a new car, you sold your old car and took a 180-day note for $4,500 at 7.5% simple interest as payment. The principal plus interest will be due at the end of 180 days.

First, let's calculate the total amount due at the end of the 180 days. To do this, we'll use the formula for simple interest:

Interest = Principal x Rate x Time
Interest = $4,500 x 7.5% x (180/360) [As it's for half a year]
Interest = $4,500 x 0.075 x 0.5
Interest = $168.75

Now, let's add the interest to the principal to find the total amount due:
Total Amount = Principal + Interest
Total Amount = $4,500 + $168.75
Total Amount = $4,668.75

Sixty days later, you sell the note to a third party for $4,550. The third party will receive the remaining interest for 120 days. We need to find the annual interest rate for the third party's investment.

First, let's find the remaining interest:
Remaining Interest = Total Amount - $4,550
Remaining Interest = $4,668.75 - $4,550
Remaining Interest = $118.75

Now, let's find the annual interest rate for the third party's investment using the simple interest formula, but solving for the rate:

Rate = (Remaining Interest) / (Principal x Time)
Rate = $118.75 / ($4,550 x (120/360))
Rate = $118.75 / ($4,550 x 0.3333)

Rate ≈ 0.077 (approximated to 3 decimal places)

So, the annual interest rate the third party will receive for the investment is approximately 7.7%.

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The amount of penny that you should receive from your friend after the seventh shoveling job would be = 2,187 pennies.

To calculate the number of penny that you will receive after the seventh job the following is carried out.

The agreement stated that the money would be tripled for each completed shoveling job.

That is for 2 jobs = 3×3 = 9

3 jobs = 3×3×3 = 27

7 jobs = 3×3×3×3×3×3×3

= 2,187 pennies.

Therefore, the 2,187 pennies would be received from your friend after the seventh shoveling job.

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A function that forms an arithmetic sequence include the following: D. F(x) = 2x - 4.

In Mathematics and Geometry, the nth term of an arithmetic sequence can be calculated by using this equation:

aₙ =  a₁ + (n - 1)d

Where:

Next, we would determine the common difference as follows.

Common difference, d = a₂ - a₁

Common difference, d = -6 + 8 = -4 + 6 = -2 + 4

Common difference, d = -2.

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There will be a total of 7 rows for nails and 6 rows for screws, making 13 rows in total.

To solve it, we need to find the greatest common divisor (GCD) of the number of boxed nails (42) and boxed screws (36). This will tell us the greatest number of boxes that can be displayed in one row without mixing nails and screws.

Step 1: List the factors of each number.
- Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Step 2: Find the greatest common divisor (GCD) by identifying the largest factor they have in common.
- The largest common factor is 6.

So, the greatest number of boxes that can be displayed in one row is 6.

Next, we'll find out how many rows will there be if the manager puts the greatest number of boxes in each row.

Step 3: Divide the total number of boxed nails and boxed screws by the GCD.
- Rows for nails: 42 ÷ 6 = 7
- Rows for screws: 36 ÷ 6 = 6

Therefore, there will be a total of 7 rows for nails and 6 rows for screws, making 13 rows in total.

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The surface area of the right rectangular prism is 880 cm².

Given Top and bottom measurements of the rectangular prism = 15 cm and 11 cm

The front and back measurements of the rectangular prism = 11cm and 11cm

The two sides measurements of the rectangular prism = 11 cm and 14 cm

To find the surface area of the rectangular prism, we have to substitute the above values in the below equation,

Surface area = (top*bottom) + (front*back) + (sides)

Surface area = (15 cm * 11 cm) + (11 cm * 11 cm) + (11 cm * 14 cm)

Surface area = 330 cm² + 242 cm²  + 308 cm²  

Surface area = 880cm²

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The possible values of x as required to be determined in the task content are; ±½.

It follows from the task content that the possible values of x are to be determined from the given task content.

The given equation can be written algebraically as;

√(8x) • √(10x) • √(3x) • √(15x) = 15

√3600x² = 15

60x² = 15

x² = 15 / 60

x² = 1/4.

x = ± ½.

Ultimately, the possible values of x as required in the task content are; +½ and -½.

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Definite integral of ∫(9x^2 - 4x - 1)dx from a to b is 3(b^3 - a^3) - 2(b^2 - a^2) - (b - a).

To evaluate the definite integral ∫(9x^2 - 4x - 1)dx, you need to first find the indefinite integral (also known as the antiderivative) of the function 9x^2 - 4x - 1. The antiderivative is found by applying the power rule of integration to each term separately:
∫(9x^2)dx = 9∫(x^2)dx = 9(x^3)/3 = 3x^3
∫(-4x)dx = -4∫(x)dx = -4(x^2)/2 = -2x^2
∫(-1)dx = -∫(1)dx = -x
Now, sum these results to obtain the antiderivative:
F(x) = 3x^3 - 2x^2 - x
∫(9x^2 - 4x - 1)dx from a to b = F(b) - F(a)

To evaluate the definite integral ∫(9x^2 - 4x - 1)dx =, we need to use the formula for integrating polynomials. Specifically, we use the power rule of integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration.
Using this formula, we integrate each term in the given expression separately. Thus, we have:
∫(9x^2 - 4x - 1)dx = (9∫x^2 dx) - (4∫x dx) - ∫1 dx
                  = 9(x^3/3) - 4(x^2/2) - x + C
                  = 3x^3 - 2x^2 - x + C
Next, we need to evaluate this definite integral. A definite integral is an integral with limits of integration, which means we need to substitute the limits into the expression we just found and subtract the result at the lower limit from the result at the upper limit. Let's say our limits are a and b, with a being the lower limit and b being the upper limit. Then, we have:
∫(9x^2 - 4x - 1)dx from a to b = [3b^3 - 2b^2 - b] - [3a^3 - 2a^2 - a]
                                              = 3(b^3 - a^3) - 2(b^2 - a^2) - (b - a)
Therefore, the definite integral of ∫(9x^2 - 4x - 1)dx from a to b is 3(b^3 - a^3) - 2(b^2 - a^2) - (b - a).

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b. <2 ≅ < 3; corresponding angles are equal

d. < 1 + < 2 = 180 degrees; sum of angles on a straight line

To determine the reasons, we need to know about transversals

Transversals are lines that passes through two lines at the given plane in two distinct points.

It intersects two parallel lines

It is important to note the following;

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The loan with the smallest monthly payment is the 30-month loan at 6% annual simple interest rate, with a monthly payment of $45.83.

To determine the loan with the smallest monthly payment, we need to calculate the monthly payment for each loan option and compare them.

We can use the formula for monthly payment on a simple interest loan:

monthly payment = (principal + (principal * interest rate * time)) / total number of payments

where:

We can compute the monthly payments for each loan choice using this formula:

1. 12 Monthly interest rate = 0.0625/12 = 0.00521, monthly payment = (1250 + (1250 * 0.00521 * 12)) / 12 = $107.35

2. 18 months at 6.75%: monthly interest rate = 0.0675/12 = 0.00563, monthly payment = (1250 + (1250 * 0.00563 * 18)) / 18 = $81.96

3. 24 months at 6.5%: monthly interest rate = 0.065/12 = 0.00542, monthly payment = (1250 + (1250 * 0.00542 * 24)) / 24 = $66.14

4. 30 months at 6%: monthly interest rate = 0.06/12 = 0.005, monthly payment = (1250 + (1250 * 0.005 * 30)) / 30 = $45.83

Based on these calculations, the loan with the smallest monthly payment is the 30-month loan at 6% annual simple interest rate, with a monthly payment of $45.83.

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The probability of spinning fewer than four nines is 1,626,101,367 / 1073741824, which simplifies to approximately 1.514%.

To find the probability of spinning fewer than four nines, we need to first calculate the total number of possible outcomes. The spinner can land on any of the eight sections on the board, and it is spun 10 times. So, the total number of possible outcomes is 8^10, which is 1073741824.

Next, we need to calculate the number of outcomes where fewer than four nines are spun. We can do this by finding the number of outcomes with 0, 1, 2, or 3 nines, and adding them up.

To find the number of outcomes with 0 nines, we need to find the number of ways to choose from the non-nine sections on the board. There are 5 non-nine sections, and we need to choose 10 of them. This is a combination problem, and the number of outcomes is 252.

To find the number of outcomes with 1, 2, or 3 nines, we need to use a similar approach. We can use combinations to find the number of ways to choose the nines and the non-nines, and then multiply them together. The number of outcomes with 1 nine is 9 x 5^9, with 2 nines is 9 x 9 x 5^8, and with 3 nines is 9 x 9 x 9 x 5^7.

Adding up all these outcomes, we get 252 + 9 x 5^9 + 9 x 9 x 5^8 + 9 x 9 x 9 x 5^7 = 1,626,101,367.

So, the probability of spinning fewer than four nines is 1,626,101,367 / 1073741824, which simplifies to approximately 1.514%.

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The differential equation that describes the situation is: dp/dt = 41.43 * p * (1 - p/555).

The logistic differential equation is a commonly used model for population growth or decay, taking into account the carrying capacity of the environment. It is given by:

dp/dt = r * p * (1 - p/K)

where p is the population, t is time, r is the growth rate, and K is the carrying capacity.

In this case, the maximum capacity of the island is 555 bears, so we have K = 555. At 6 AM, the number of bears on the island is 165 and is increasing at a rate of 29 bears per day, so we have:

p(0) = 165 and dp/dt(0) = 29

To write the differential equation that describes this situation, we can use the initial conditions and the logistic model:

dp/dt = r * p * (1 - p/555)

Substituting the initial conditions, we get:

29 = r * 165 * (1 - 165/555)

Simplifying this expression, we get:

29 = r * 0.7

r = 41.43

Therefore, the differential equation that describes the situation is:

dp/dt = 41.43 * p * (1 - p/555)

Note that this model assumes that the growth rate of the bear population is proportional to the number of bears present and that the carrying capacity is fixed. Real-life situations may involve more complex models with time-varying carrying capacities or other factors affecting population growth.

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Alfonso received a 30% overall discount on the refrigerator and the stove together if the sale price of the refrigerator was 2/3 of the sale price of the stove. Therefore, statement (c) is the correct answer.

To see why, let's use some algebra. Let R be the original price of the refrigerator and S be the original price of the stove. Then the sale price of the refrigerator is 0.6R and the sale price of the stove is 0.8S. The total amount Alfonso paid is 0.6R + 0.8S.

To receive a 30% overall discount, Alfonso must have paid only 0.7 times the total original price, which is 0.7(R + S). Therefore, we have the equation:
0.6R + 0.8S = 0.7(R + S)

Simplifying this equation gives:
0.1R = 0.1S
R = S

So the original prices of the refrigerator and the stove were the same. This means that statement (C) is  true.

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Complete Question:  

Alfonso went to famous Sam's Appliances store and purchased a refrigerator and a stove. The sale price of the refrigerator was 40% off the original price and the sale price of the stove was 20% off the original price.

Which statement must be true to conclude that Alfonso received a 30% overall discount on the refrigerator and the stove together?

a) The sale prices of the refrigerator and the stove were the same.

b) The original prices of the refrigerator and the stove were the same.

c) The sale price of the refrigerator was twice the sale price of the stove.

d) The original price of the refrigerator was twice the original price of the stove.

The volume of the diamond is  609.3 cm³.

To find the volume of the diamond, we can use the formula:

Volume = Mass / Density

Given:

Mass = 2138.7 grams

Density = 3.51 g/cm³

Substituting these values into the formula:

Volume = 2138.7 g / 3.51 g/cm³

Calculating the division:

Volume ≈ 609.3 cm³

Therefore, the volume of the diamond is approximately 609.3 cm³.

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Answer:

238 ft²

Step-by-step explanation:

Base area is l*w and there are two of these so 2(7*7) = 98

Then there 4 faces that have dimensions 4(7*5)=140

98+140=238

The surface area of the cone is approximately 793.10 cm².

What is cone?

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.

What is area of cone?

The surface area of a cone is given by the formula:

Surface Area = πr² + πrℓ

where r is the radius of the circular base and ℓ is the slant height of the cone.

To find the surface area of a cone, we need to find the area of its base and the area of its lateral surface and then add them together.

The formula for the lateral surface area of a cone is given by:

L = πrℓ

where r is the radius of the base and ℓ is the slant height of the cone.

The formula for the area of the base of a cone is given by:

B = πr²

where r is the radius of the base.

Given the slant height ℓ = 19 and the radius r = 9, we can find the lateral surface area of the cone as follows:

L = πrℓ

= π(9)(19)

≈ 538.63 cm²

Next, we can find the area of the base of the cone as follows:

B = πr²

= π(9)²

≈ 254.47 cm²

Therefore, the surface area of the cone is:

A = B + L

≈ 793.10 cm²

So, the surface area of the cone is approximately 793.10 cm².

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Step-by-step explanation:

Finding the square root of a number is simply jist dividing the given number by 2 till you get to the last number which should be 1 then you pair up the number of two's and multiply. For example you have 6 two's when you pair them in two's you get 3 two's left then you multiply the remaining two's which would then be 2×2×2 which is 6

Answer:

Step-by-step explanation:

so basically, a square root is just the number that is multipled together to equal that, so sq rt of 64 is 8. This is because 8x8= 64. If you were to take a weird number like sq rt of 112, it would be a little bit more difficult, however its pretty easy to do this through thinking of your multiplication charts. if you think about it 11x10 but thats 110, it would have to be a number around there. 11x11 is too high but 10x10 is too low. SO it would have to be a number around there. if you did 10.5x10.5 it would give 110.25. so if you try to do 10.6x10.6 it would equal 112.36. (The actual answer is 10.58300524425836)  So that is how you determine how close you can get. It is very tedious to do this process and very time consuming. However, i would just advise you try to use a calculator (??)

this may be helpful if the videos aren't working for you :))

The value of h in the triangle shown above is calculated using proportion as: h = 4.

The two triangles shown are similar to each other based on the Angle-angle (AA) Similarity theorem. This implies that the length of their corresponding pair of sides would be proportional to each other.

Therefore, we have:

8/18 = h/9 [proportional sides of similar triangles]

Cross multiply:

h * 18 = 8 * 9

18h = 72

Divide both sides by 18:

18h/18 = 72/18 [Division property of equality]

h = 4

Therefore, the length of h in the given image is determined as: 4 units.

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Kareem drove the truck for 150 miles.

As areem rented a truck for one day. there was a base fee of $19.99 , and there was an additional charge of 83 cents for each mile driven we will let the number of miles driven by Kareem be represented by 'x'. The total cost, including the base fee, can be represented as:

Total cost = Base fee + Additional charge per mile * Number of miles driven

$135.36 = $19.99 + $0.83x

Subtracting $19.99 from both sides and then dividing by $0.83, we get:

x = (135.36 - 19.99) / 0.83

x = 150

Therefore, Kareem drove the truck for 150 miles.

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The simplification of the given algebraic expression is: -3x - 9y

Algebraic expressions are defined as the idea of expressing numbers with the aid of letters or alphabets without really specifying their actual values.

Now, the algebraic operations are known by the acronym PEMDAS which denotes:

P- Parentheses, E- Exponents, M- Multiplication, D- Division, A- Addition, and S- Subtraction.

We want to find the sum of the algebraic expressions given as (-10x - 4y) and (7x - 5y) in simplest form.

Thus, we have:

(-10x - 4y) + (7x - 5y)

= -10x - 4y + 7x - 5y

= -3x - 9y

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Complete question:

Find an expression which represents the sum of (-10x - 4y) and (7x - 5y) in simplest terms.

the series converges for all x in the interval (-1/6, 1/6) in interval notation.

The given series is a geometric series with first term a=1 and common ratio r=6x. The series converges if and only if |r|<1.

So, |6x|<1

Solving this inequality, we get:

-1/6 < x < 1/6

To find all values of x for which the series converges, we need to analyze the given series:
∑(6x)^n

This is a geometric series with a common ratio of 6x. For a geometric series to converge, the absolute value of the common ratio must be less than 1:

|6x| < 1

Now, we can solve for the interval of x:

|-1/6| < |x| < |1/6|

Using interval notation, the range of values for which the series converges is:

(-1/6, 1/6)

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Acellus is a powerful tool for students looking to achieve academic success and gain the skills they need to succeed in today's competitive world.

Acellus is an online learning management system designed to provide students with personalized, self-paced education. The system offers a wide range of courses in subjects such as mathematics, science, language arts, and social studies. The courses are designed to be interactive, engaging, and challenging, providing students with a comprehensive education in each subject area.

Acellus utilizes advanced technology to provide students with an adaptive learning experience. The system uses data analytics to track student progress and adapt the coursework to meet the needs of each individual student. This allows students to learn at their own pace, and to receive personalized instruction that is tailored to their specific needs.

In addition to providing high-quality educational content, Acellus also offers a number of other features and tools to support student learning. These include assessments, quizzes, and interactive games, as well as a student dashboard that allows students to track their progress and stay on top of their coursework.

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Answer:

Step-by-step explanation:

In this carnival game, a player flips a coin that has a 3 on one side and a 7 on the other, and then rolls a number cube that has numbers 1-6.

To determine the probabilities, we need to analyze each event separately and then use the multiplication rule of probability to find the probability of both events happening together.

The probability of getting a 3 on the coin is 50%, since there are only two possible outcomes. The probability of rolling each number on the cube is 16.67%, since the cube has six sides.

The probability of both events happening together depends on the individual probabilities and is found by multiplying them. Finally, we can use the addition rule of probability to find the probability of either event happening.

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"A recipe for chocolate chip cookies calls for 4/5 cup of sugar per batch. If a baker wants to make 3 batches of cookies, and only has 1/4 cup of sugar left in the pantry, how much additional sugar will the baker need to buy?" is an example of a word problem for the given equation.

To solve this word problem, we can use the equation 4/5 x 1/4 + 3/5 = to find out how much sugar is needed for one batch of cookies, and then multiply that amount by 3 to get the total amount of sugar needed for 3 batches.

The first part of the equation, 4/5 x 1/4, represents the amount of sugar needed for one batch of cookies, which is 1/5 cup. Adding the remaining 3/5 cup of sugar needed for the recipe gives a total of 4/5 cup of sugar per batch.

To find out how much additional sugar the baker needs to buy, we can multiply 4/5 by 3 (the number of batches), and then subtract the amount of sugar already in the pantry (1/4 cup). This gives us:

4/5 x 3 - 1/4 = 12/5 - 1/4 = 43/20

Therefore, the baker will need to buy 43/20 cups of additional sugar to make 3 batches of chocolate chip cookies.

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The trigonometric function equivalent to sec(-270) is -1.

The secant function is defined as the reciprocal of the cosine function, i.e., sec(x) = 1/cos(x). To find the value of sec(-270), we need to first find the cosine of -270 degrees. The cosine function has a period of 360 degrees, which means that cos(-270) is the same as cos(-270 + 360) = cos(90) = 0. Therefore, we have sec(-270) = 1/0, which is undefined.

However, we can determine the sign of sec(-270) by examining the quadrant in which the angle -270 degrees lies. Since -270 degrees is in the fourth quadrant, the cosine function is negative in that quadrant. Therefore, we can write sec(-270) = -1/0-, which is equivalent to -1. Hence, the trigonometric function equivalent to sec(-270) is -1.

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To find the projection of u onto v, we need to use the formula:

projv(u) = (u · v / |v|^2) * v

where u · v is the dot product of u and v, and |v|^2 is the magnitude of v squared.

First, we calculate u · v:

u · v = (-8i - 4j - 2k) · (0i + 2j + 2k) = 0 + (-8*0) + (-4*2) + (-2*2) = -8

Next, we calculate |v|^2:

|v|^2 = (2)^2 + (2)^2 = 8

Now, we can plug these values into the projection formula:

projv(u) = (-8 / 8) * (2j + 2k) = -1j - 1k

So the projection of u onto v is -1j - 1k.

To find the vector component of u orthogonal to v, we can use the formula:

u - projv(u)

We already found projv(u) to be -1j - 1k, so we just need to subtract that from u:

u - projv(u) = (-8i - 4j - 2k) - (-1j - 1k) = -8i - 3j - k

Therefore, the vector component of u orthogonal to v is -8i - 3j - k.
To find the projection of u onto v, we will use the formula:

proj_v(u) = (u•v / ||v||^2) * v

where u = -8i - 4j - 2k, v = 2j + 2k, and • denotes the dot product.

First, calculate the dot product (u•v):

u•v = (-8)(0) + (-4)(2) + (-2)(2) = -8 - 4 = -12

Next, calculate the magnitude squared of v (||v||^2):

||v||^2 = (0^2) + (2^2) + (2^2) = 0 + 4 + 4 = 8

Now, use the formula:

proj_v(u) = (-12 / 8) * v = -1.5 * (2j + 2k) = -3j - 3k

Next, to find the vector component of u orthogonal to v, use the formula:

u_orthogonal = u - proj_v(u)

u_orthogonal = (-8i - 4j - 2k) - (-3j - 3k) = -8i - 1j + 1k

So, the projection of u onto v is -3j - 3k, and the vector component of u orthogonal to v is -8i - 1j + 1k.

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The density of the copper used in the pyramid is 8.4 g/cm³.

To find the density of the copper used in the pyramid, we first need to determine the volume of the cuboid and pyramid, and then find the mass of the copper.

1. Find the volume of the cuboid (V_cuboid):

V_cuboid = length × width × height

Since the base is a square, the length and width are both 3 cm.

V_cuboid = 3 cm × 3 cm × 8 cm = 72 cm³

2. Find the volume of the pyramid (V_pyramid):

First, find the height of the pyramid: total height (15 cm) - height of the cuboid (8 cm) = 7 cm.

V_pyramid = (1/3) × area of base × perpendicular height

The area of the base is 3 cm × 3 cm = 9 cm².

V_pyramid = (1/3) × 9 cm² × 7 cm = 21 cm³

3. Find the mass of the iron cuboid (m_iron):

Density of iron = 7.8 g/cm³

m_iron = density × V_cuboid = 7.8 g/cm³ × 72 cm³ = 561.6 g

4. Find the mass of the copper pyramid (m_copper):

Total mass of sculpture = 738 g

m_copper = total mass - m_iron = 738 g - 561.6 g = 176.4 g

5. Calculate the density of the copper (density_copper):

density_copper = m_copper / V_pyramid

density_copper = 176.4 g / 21 cm³ ≈ 8.4 g/cm³

The density of the copper is approximately 8.4 g/cm³.

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The Number of games lost cannot be negative .

The number of games the Senators won as "W." According to the given information, the Senators won 118 more games than they lost. This can be expressed as:

W = L + 118,

where "L" represents the number of games the Senators lost.

the Senators played a total of 78 games, which means the number of games they won and lost combined should equal 78:

W + L = 78.

Substituting the first equation into the second equation, we have:

(L + 118) + L = 78,

2L + 118 = 78,

2L = 78 - 118,

2L = -40,

L = -40/2,

L = -20.

Since the number of games lost cannot be negative.

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Answer:

A dilation always produces a similar figure. Similar figures have the same shape but different sizes.

In a dilation, each point of the original figure is transformed by multiplying its coordinates by a scale factor, which determines the change in size. However, the shape and proportions of the figure remain unchanged. Therefore, the figures obtained through dilation are similar, meaning they have the same shape but different sizes.

The given expression is sin⁻¹ (sin((-5t-6)). Since the argument of sin⁻¹ and sin is the same, we can simplify the expression as follows:
sin⁻¹ (sin((-5t-6))) = -5t-6
OR, -5t-6 = (-2π/π)(-5t-6/2) = -2π(2.5t+3)/π = -5π/2(2.5t+3)
Therefore, the answer is -5π/2(2.5t+3).

Given the expression: sin^-1(sin(-5t-6))
To find the angle -z/2, we can use the following properties:
1. sin⁻¹ (sin(x)) = x, if -π/2 ≤ x ≤ π/2 (i.e., x is in the range of the principal branch of the inverse sine function).
2. The sine function has a periodicity of 2π. Therefore, sin(x) = sin(x + 2nπ), where n is an integer.

Given angle: -5t - 6
We need to add 2nπ to this angle to bring it into the range of -π/2 to π/2:
⇒ -5t - 6 + 2nπ, where n is an integer.

Now, we apply the sine and inverse sine functions:
sin⁻¹ (sin(-5t - 6 + 2nπ))

Since sin^-1(sin(x)) = x when x is in the range of the principal branch, our final expression becomes:
-z/2 = -5t - 6 + 2nπ

In this expression, -z/2 represents the angle in radians, written as a multiple of r. To find the multiple, you simply have to solve for -z/2 in terms of r.
Therefore, the answer is: -z/2 = -5t - 6 + 2nπ.

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